Cell decompositions of moduli space, lattice points and Hurwitz problems

Abstract

In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes which come equipped with natural integer points and a volume form. We show how to effectively calculate the number of lattice points and the volumes over all the cells and that these calculations contain deep information about the moduli space.

38 pages, To appear in Handbook of Moduli

References in corpus (1)

• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

Cited by in corpus (2)

• Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces
• Counting lattice points in compactified moduli spaces of curves